Exam-Style Questions on Differentiation Optimisation

A child's play tent is made in the shape of half a cylinder. It is constructed from a fibreglass frame with material pulled tightly around it. The fibreglass frame consists of a rectangular base, two semi-circular ends and two further support rods, as shown in the following diagram.

The semicircular ends each have radius \(r\) and the support rods each have length \(d\).

Let F be the total length of fibreglass used in the frame of the play tent.

(a) Write down an expression for \(F\) in terms of \(r\), \(d\) and \(\pi\).

The volume of the play tent is 0.95 m³.

(b) Write down an equation for the volume of the play tent in terms of \(r\), \(d\) and \(\pi\).

(c) Show that \(F = 2\pi r + 4r + \frac{7.6}{\pi r^2}\)

(d) Find \(\frac{dF}{dr}\)

The play tent is designed so that the length of fibreglass used in its frame is a minimum.

(e) Find the value of \(r\) for which \(F\) is a minimum.

(f) Calculate the value of \(d\) for which \(F\) is a minimum.

(g) Calculate the minimum value of \(F\).

Worked Solution

A package is in the shape of a cuboid and has a length \(l\) cm, width \(w\) cm and height of 12 cm.

(a) Express the volume of the package in terms of \(l\) and \(w\).

The total volume of the package is 2400 cm³.

(b) Show that \(l=\frac{200}{w}\).

The package is tied up using a length of red string that fits exactly around the package in two different directions, as shown in the following diagram (not to scale).

(c) Show that the length of string, \(x\)cm, required to tie up the package can be written as \(24+4w+\frac{400}{w}\)

(d) Sketch the graph of \(x\) for \(0\lt w \le 12\), clearly showing the local minimum point.

(e) Find \(\frac{dx}{dw}\).

(f) Find the value of \(w\) for which \(x\) is a minimum.

(g) Find the value, \(l\), of the package for which the length of string is a minimum.

(h) Find the minimum length of string required to tie up the package.

Worked Solution